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In prior Carnivals of Mathematics it has been a tradition to include trivia about the number the Carnival happens to be at in some way. With a desire to do something different for Math Teachers at Play, I offer a riddle. Younger solvers may have an advantage over older ones here.

Here is a pair of dice I own:

If I roll the dice and read the numbers off the top, is it more likely the numbers add up to be 2, or add up to be 12? Or are the two sums equally likely? Why?

Finally, this is a series that appeared back in 2008, but it was new to me, so I hope it is new to some of you! Ron Doerfler wrote a three-part series on “lightning calculators”, people who can do astounding mathematical calculations in their head: The Players, The Methods, The Media. Especially fun (in part 3) is the deconstruction of a Daniel Tammet documentary and its “creative” editing.

I’d already figured the picture was important, but I can’t discern anything significant from it. The dice appear to be normal, fair, and equal, to the extent that one can tell from a blurry photo.

We can’t see the 4 and 1 sides, so perhaps there’s some trick to the dice involving that. Like, maybe there’s no 1 opposite the 6, so a roll of 2 is impossible. But that’d be cheesy, because there’s no real clue of any such thing.

For the riddle: is it that we know that there is a six on each die, but we don’t know whether or not there is a one on each die (because that side is not pictured, and conceivably they could be non-standard dice), so the probability of getting 12 is higher?

It’s hard to tell, but from the picture I think I notice a slight concavity in the material used to backfill the holes (12 of them for boxcars). So it is conceivable that there is overall less mass on the side with a ‘6’ than there is on the side with a ‘1’, which would lend a small bias in favor of the dice landing with the 1’s on the bottom and 6’s on top.

Surely the difference is so minor as to be far overshadowed by other factors (such as other minuscule irregularities in the dice). This seems rather like the specious claim that drains empty in different directions south of the equator vs north, owing to the Coriolis effect.

Have you actually done an experiment to show that there’s any real statistical difference?

Yes, I have witnessed the effect for myself. It is also well known in the gambling literature. (That’s where I first came across it — I think it may have been Scarne on Dice.)

The weight effect is generally far greater than any manufacture error in the dice (relatively speaking, as I said in the other post, you need at least 500 rolls before you have a chance of spotting anything), although it is a good point that cheap enough quality might override everything.

Actually, there is the same exact odds of rolling 2 or 12 each and everytime the dice are thrown. The odds of rolling each are 1:36. This is why at a casino, the payout when you bet on either is 1:30; Sounds like great odds, but you are still the underdog by the odds of 1:6. Which means statistically, the casino has the better odds of winning by 6:5 overall, or for each roll. or 36:30.

Sir Chadwick, you make me wonder whether you even read the comments above. The dice pictured above are cheaply produced dice, not high-quality casino dice. With cheaply produced dice, holes are drilled and not too carefully backfilled, and this definitely introduces a bias. Statisticians familiar with both kinds of dice (such as Mosteller) are quite familiar with the phenomenon.

[…] Teachers at Play #29 Posted on August 23, 2010 by Jason Dyer The last time I hosted Math Teachers at Play I attempted to start a tradition of including a math puzzle pertinent to the number of the […]